A rigid rocket of proper length (its length measured in the frame in which it is…

PHYSICS · Multi-part · Fermi University Physics · Relativity & Gravitation · The equivalence principle and gravity as geometry · The equivalence principle and gravity as geometry

Relativity and Gravitation, Ch Accelerated observers, hyperbolic motion and the equivalence principle

Problem

A rigid rocket of proper length

500\ \mathrm{m}

(its length measured in the frame in which it is momentarily at rest) carries a clock at the nose and a clock at the tail. Starting from rest at

t = 0

in the inertial launch frame

S

, it burns with constant proper acceleration

a = 25.0\ \mathrm{m\,s^{-2}}

along the

+x

axis, and the tail clock records a proper time

\tau = 1.00\times10^{7}\ \mathrm{s}

of burn. Take

c = 2.998\times10^{8}\ \mathrm{m\,s^{-1}}

G = 6.674\times10^{-11}\ \mathrm{N\,m^{2}\,kg^{-2}}

, and for the Earth

M = 5.972\times10^{24}\ \mathrm{kg}

, radius

6.37\times10^{6}\ \mathrm{m}

. All clock-rate results are wanted to first order in

ah/c^{2}

, and any gravitational field is ignored for the rocket parts. ((a)) At the end of the burn, find the rocket's speed in

S

as a multiple of

c

, and the time elapsed in

S

since launch. ((b)) What is the rocket's length in

S

at that instant? ((c)) By how much has the nose clock got ahead of the tail clock over the burn? ((d)) The crew argue that, by the Einstein equivalence principle, their cabin is locally like a laboratory at rest on the Earth's surface. To test the locality condition they release two marbles at rest,

500\ \mathrm{m}

apart along the direction of acceleration; in the rocket their relative acceleration is zero. What would the magnitude of the relative acceleration of two such marbles be, released

500\ \mathrm{m}

apart along the vertical just outside the Earth's surface?

Browse the problem library